Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality

TL;DR

This paper develops new inequalities for $L^p$ norms that improve the triangle inequality, interpolating between disjoint and overlapping functions, and are valid for all $p$, thus extending and strengthening previous results.

Contribution

The authors establish stronger inequalities for $L^p$ norms that interpolate between known bounds, generalizing Carbery's question and extending the validity to all $p$ values.

Findings

New inequalities for $L^p$ norms that sharpen the triangle inequality.

Interpolation between disjoint and overlapping functions using $ orm{fg}_{p/2}$.

Results are valid for all $p$, not just specific cases.

Abstract

In 2006 Carbery raised a question about an improvement on the na\"ive norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carbery's question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $p$.